Let $(X, \mathcal{U})$ be a uniform space and $C\in\mathcal{U}$ be given. Is there a compact set $D\in\mathcal{U}$ such that $D\subseteq C$?

Question

Let $X$ be first countable, locally compact, paracompact, Hausdorff space. We know that $X$ has a uniformity $\mathcal{U}$. Thus $(X, \mathcal{U})$ is a uniform space.

Is it true that :

For every $E\in\mathcal{U}$, there is compact set $D\in\mathcal{U}$ such that $D\subseteq E$.

Please help me to know it.

Answer 1

If such a $D$ existed, $\Delta_X$ would be a closed subset of it, and so $X$ would be compact as $\Delta_X \simeq X$. So this only happens in the trivial case that $X$ is compact Hausdorff.